
If the linear system (REF ) is consistent, i.e., \(=_{\textsc {r}}\)  or \(_{\textsc {n}}=0\) , then it may has infinitely
many solutions. As shown in [1]}, the solution \(^{\ast } = A^\dag \)  is just the least Euclidean-norm solution, which satisfies

 = n, s.t. A=.

From the Petrov-Galerkin conditions, we know that \(_k^T(- A^{(k+1)})=0\) . By the Pythagorean theorem and \(=_{\textsc {r}}=AA^\dag \) , for \(k\ge 1\) , we have

(k+1) - A†= (k) - A† -
| kT(- A(k)) |2AT k.

Since

kT(- A(k))
 = ikUk (bik - ikT (k)) ikT (- A(k))
